![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > scutfo | Structured version Visualization version GIF version |
Description: The surreal cut function is onto. (Contributed by Scott Fenton, 23-Aug-2024.) |
Ref | Expression |
---|---|
scutfo | ⊢ |s : <<s –onto→ No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scutf 27866 | . 2 ⊢ |s : <<s ⟶ No | |
2 | lltropt 27920 | . . . . 5 ⊢ ( L ‘𝑥) <<s ( R ‘𝑥) | |
3 | df-br 5170 | . . . . 5 ⊢ (( L ‘𝑥) <<s ( R ‘𝑥) ↔ 〈( L ‘𝑥), ( R ‘𝑥)〉 ∈ <<s ) | |
4 | 2, 3 | mpbi 230 | . . . 4 ⊢ 〈( L ‘𝑥), ( R ‘𝑥)〉 ∈ <<s |
5 | lrcut 27950 | . . . . 5 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥) | |
6 | 5 | eqcomd 2740 | . . . 4 ⊢ (𝑥 ∈ No → 𝑥 = (( L ‘𝑥) |s ( R ‘𝑥))) |
7 | fveq2 6919 | . . . . . 6 ⊢ (𝑦 = 〈( L ‘𝑥), ( R ‘𝑥)〉 → ( |s ‘𝑦) = ( |s ‘〈( L ‘𝑥), ( R ‘𝑥)〉)) | |
8 | df-ov 7448 | . . . . . 6 ⊢ (( L ‘𝑥) |s ( R ‘𝑥)) = ( |s ‘〈( L ‘𝑥), ( R ‘𝑥)〉) | |
9 | 7, 8 | eqtr4di 2792 | . . . . 5 ⊢ (𝑦 = 〈( L ‘𝑥), ( R ‘𝑥)〉 → ( |s ‘𝑦) = (( L ‘𝑥) |s ( R ‘𝑥))) |
10 | 9 | rspceeqv 3653 | . . . 4 ⊢ ((〈( L ‘𝑥), ( R ‘𝑥)〉 ∈ <<s ∧ 𝑥 = (( L ‘𝑥) |s ( R ‘𝑥))) → ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦)) |
11 | 4, 6, 10 | sylancr 586 | . . 3 ⊢ (𝑥 ∈ No → ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦)) |
12 | 11 | rgen 3065 | . 2 ⊢ ∀𝑥 ∈ No ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦) |
13 | dffo3 7134 | . 2 ⊢ ( |s : <<s –onto→ No ↔ ( |s : <<s ⟶ No ∧ ∀𝑥 ∈ No ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦))) | |
14 | 1, 12, 13 | mpbir2an 710 | 1 ⊢ |s : <<s –onto→ No |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2103 ∀wral 3063 ∃wrex 3072 〈cop 4654 class class class wbr 5169 ⟶wf 6568 –onto→wfo 6570 ‘cfv 6572 (class class class)co 7445 No csur 27693 <<s csslt 27834 |s cscut 27836 L cleft 27893 R cright 27894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-1o 8518 df-2o 8519 df-no 27696 df-slt 27697 df-bday 27698 df-sslt 27835 df-scut 27837 df-made 27895 df-old 27896 df-left 27898 df-right 27899 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |